Regions are Shaded Weirdly
As said above, in a comment, you should at least 
- in edit mode
- check doubles W and choose "remove doubles"
- recalculate normals CTRLN

and in general check that you don't have duplicated geometry.

about the weird shading in the bottom faces, this is what happens, and how to solve (using a simple cylinder):



what happens is the effect of the bottom face being distorted by the subdivision surface modifier, without having some more supporting geometry to calculate a better shape for that 90° angle between the bottom and the sides.

here is this problem shown as flat shading



and here with smooth shading: less ugly but still visible.



To fix this weird bottom shades, we need to add some supporting geometry. 
In edit mode, add a loop CTRLR



and slide it down almost to the bottom, confirm.



Now we need to do something similar for the bottom (n-gon) face.



Select it and delete (only faces)



now select this edge circle and extrude E, then press esc to stop the extrusion, 



and instead scale it towards the center, using S



now extrude the inner circle (selected) again towards the center



and lastly close the inner small circle left open with F: it will create a small n-gon, but it won't create issues here. It could be converted to triangles or just quads, if needed.



Now, back into object mode, the subsurf is no more creating those artifact, since ther is now enough geometry on that corner to allow for a nice, uifomr smoothing.